Three Frames

It is a straightforward exercise in coordinate geometry to model the Earth orbiting around the Sun.

To model the Solar System, every celestial body needs a handful of parameters (size, orbital speed, orbital radius, etc.). As we add planets, moons, and even potentially moons of moons, the orbital trajectories become more complicated. Even though each trajectory is a smaller body revolving around a larger body, the equations get more complicated at each step. The problem: can we frame the problem in such a way that everything uses relative instead of absolute coordinates?

This app demonstrates one solution to the problem that uses three frames of reference. These three frames allow us to define matrix transformations, and these transformations allow us to represent all motion and rotations using relative coordinates. If a moon is orbiting a planet, for example, then the moon’s motion can be described from the point of view of an observer on the moving planet.

More generally, the three-frame algorithm works in any scenario where there are many observers, frames of reference, or different coordinate sets. If one observer (for example the camera or a stationary body like the Sun) can be considered the master point of view, then it becomes simple to switch back and forth between any two observers’ preferred coordinates.

In this app, planets and moons can be added or removed, and their parameters can be played with using the sliders. Since everything uses relative coordinates, the trajectories are computed almost instantaneously.